Introduction to spectroscopy Applications of X-ray and neutron scattering in biology, chemistry and physics 23/ Niels Bech Christensen Department of Physics Technical University of Denmark ħωħω Q

Outline Recap of useful concepts Problems to tackle with spectroscopic tools Time and energy scales Spectroscopic instrument types and principles

Properties of the neutron Mass : kg Charge : 0 (no Coulomb) 1 0 n  1 0 p + e - + μ e Lifetime: 886(1) sekunder Spin : ½ (magnetic) Magnetic dipole moment: μ = μ N Nobel price 1935 James Chadwick

Interactions with matter 1.Nuclear interaction with atomic nuclei 2.Magnetic interaction with the angular momenta of electrons in unfilled shells Short range : ~ – m Longer range : ~ m Isotropic, central potential V n =V n (r) Anisotropic, non-central potential V m =V m ( r ) Typical magnitude of cross-section: m 2

Conservation laws Energy Momentum kiki kfkf Q Scattering triangle SpinPolarization analysis Spin echo

1994: Nobel price winners Clifford G. Shull ( ), USA "for the development of the neutron diffraction technique" Bertram N. Brockhouse ( ), Canada "for the development of neutron spectroscopy" "for pioneering contributions to the development of neutron scattering techniques for studies of condensed matter" ”Where atoms/spins are … ”… and what atoms/spins do”

Nuclear coherent and incoherent Incoherent scattering cross-section Coherent scattering cross-section (Bravais lattice) Averages over isotopes and total spin states

Nuclear coherent and incoherent Incoherent scattering cross-section Coherent scattering cross-section (Bravais lattice) Fourier transforms in temporal and spatial coordinates of 2-point correlation functions (same for magnetic interactions) Coherent scattering : Different particles (j,j sum. Interference) Incoherent scattering : The same particle (j sum. No interference)

Real and reciprocal space: Fourier transformation

Examples of Fourier transforms Point-like object  Constant in Q e.g. absence of formfactor for scattering from nucleus

Examples of Fourier transforms Atoms in a crystal  Bragg reflections (for time-independent two-point correlations)

Examples of Fourier transforms An extended object in R  An extended object in Q e.g. formfactor for magnetic scattering

Examples of Fourier transforms Double width in R  Width in Q decreases by factor 2

Time and energy: Fourier transformation

Examples Time-independent phenomena (eigenstates)  Bragg peaks, sharp inelastic modes

Examples A broad distribution of time-scales  Finite width in energy

Examples Double the width in time  Reduce the energy width by factor 2 e.g. lifetime broadening/narrowing

Outline Recap of useful concepts Problems to tackle with spectroscopic tools Time and energy scales Spectroscopic instrument types and principles

Problems to tackle by spectroscopic methods Diffraction allows us to obtain the structure of matter (crystals, polymers, etc) but that structure is itself determined by the manner in which the constituent atoms/molecules bind, and by the strength of these couplings Spectroscopy allows us to extract the value of the couplings from inelastic scattering, e.g. relating to the stiffness of a material, its magnetic properties etc. Spectroscopy also allows us to extract e.g. the rates of diffusion etc from quasielastic scattering Why do we need these values? Ultimately, because we want to be able to control the properties of future materials, which depend on the couplings. Materials design, drug development, ….

Problems to tackle by spectroscopic methods Lattice vibrations : thermoelectrics, conventional superconductors, … Magnetic excitations : magnetoelectrics, magnetocalorics, colossal magnetoresistance, unconventional superconductors,… Fundamental physical problems : (quantum) phase transitions, skyrmionics, spin ice, glassy dynamics, … Diffusion processes : e.g. of water in matter, hydrogen storage, … Vibrational spectroscopy : chemistry, catalysis, Biology and soft matter : proteins, DNA, polymers, medicine, …

Outline Recap of useful concepts Problems to tackle with spectroscopic tools Time and energy scales Spectroscopic instrument types and principles

Time and energy scales t υ 1 meV = Joule 1 THz = 4.14 meV 1 cm -1 = meV E 1 meV 4 ps 1 eV 250 THz 4 fs4 ns 250 GHz 1 μeV 250 MHz 1 neV 250 kHz 4 μs

Neutrons : r and t coverage Neutron scattering covers a very large range of (r,t) and (Q,E) including ranges not covered by complementary techniques

Types of scattering Quasi-elastic Inelastic Elastic Different spectroscopic problems (characteristic times!) impose different requirements in terms of resolution. Therefore, several spectroscopic instrument types exist

Outline Recap of useful concepts Problems to tackle with spectroscopic tools Time and energy scales Spectroscopic instrument types and principles

Instrument types: Triple-axis spectrometers Time-of-flight spectrometers (direct) Time-of-flight spectrometers (indirect) Spin echo spectrometers

Triple-axis spectrometers Key optical element: The monochromator/analyzer Bragg’s law Laue condition A good monochromator material should: Should have a high σ coh / σ inc ratio Should have a high peak reflectivity Should not be too perfect (due to extinction, backside does not scatter)

Triple-axis spectrometers Extremely simple principle due to Brockhouse 3 axes Monochromator-Sample (a1,a2=2 θ M ) Sample-Analyser (a3,a4 =2 θ s ) Analyser-Detector (a5,a6= 2 θ A ) The setting of a2 and a1=a2/2 defines k i The setting of a6 and a5=a6/2 defines k f The setting of a3 and a4 defines the Q of the crystal probed a1,a3,a5 are rotations of the mono,sample and analyser about the vertical That’s all folks! Well … not quite. Stay tuned

Triple-axis spectrometers Analyzer (Bragg reflection) Monochromator (Bragg reflection) Bragg’s law d λλ θθ Sample S(Q, ω ) High incoming flux: O(10 7 ) n/s/cm2 (IN14) Flexible Point-by-point measurements in (Q, ħω ) space Polarization analysis Cold: 2 meV <E i <15 meV Thermal: 15 meV< E i <100 meV Hot: E i >100 meV

Time-of-flight spectrometers Moderator Distance L General principles: Pulsed beams (spallation sources or chopped beams at reactors) Energies and momenta are deduced from measured time- differences (from pulse creation to neutron detection) and known distances and angles

Direct time-of-flight spectrometers MAPS, ISIS Choppers Low incident flux Low flexibility (TOF parabola) Massive parallel data acquisition in ( Q, ħ ω ) “Making maps of magnetism” Rb 2 MnF 4 Huberman et al, PRB 72, (2005) k i fixed

Indirect time-of-flight spectrometers From E f ~ λ f -2 and Bragg’s law, n λ f =2dsin( θ A ) White incident beam Bragg reflection at analysers; 2 θ A ~180° Cs 2 CuCl 4 Coldea et al, PRB 68, (2003) See Heloisa Bordallos talk for more on high resolution measurements

Neutron spin echo Principle: A magnetic moment μ in a magnetic B field precesses around B at the Larmor (angular) frequency In the time (L/v 1 ) it takes to pass the first field region, it precesses by the angle (L/v 1 ) ω L. After scattering, it enters a field region with opposite field and precesses back by -(L/v 2 ) ω L. For elastic scattering, the total phase shift will be zero. Finite phase shifts means finite energy transfer Assume v 2 = v 1 +dv (quasielastic scattering) Total phase change:

Neutron spin echo Measured quantity : Beam polarization along x (initial polarization along y). This is proportial to the cos( ωτ NSE ) Fourier transform of – essentially – the scattering cross-section. See Peter Schurtenbergers talk Because velocity is encoded in the spin- precession, one can have a broad incident flux distribution, while retaining excellent energy-resolution down to neV

Conclusions A full understanding of the properties of matter – a prerequisite for control - requires us to know both structural and dynamic properties. Dynamics can be studied using spectroscopic methods at various instrument types which probe ”motion” on different time and energy scales and with different energy-resolution conditions Coherent, propagating modes (TAS and direct TOF spectrometers) Quasi-elastic scattering (indirect TOF spectrometers and spin-echo spectrometers)

Kinematic constraints kfkf kiki Q 2θ2θ (k i fixed) (k f fixed) k f fixed k i fixed

Elastic scattering “…where atoms/spins are…” Crystal structures, polymers, biology, … Magnetic structures, flux-line lattices, … Charge and orbital ordering phenomena (Indirectly, through the associated small distortions of the lattice) … as a function of pressure, electric/magnetic field and temperature. k i =k f Q kfkf kiki ħω =0

Inelastic scattering “…what atoms/spins do…” Phonons in crystalline materials Magnetic excitations … as a function of pressure, electric/magnetic field and temperature. Q kfkf kiki Q kfkf kiki k i <k f k i >k f ħω >0 ħω <0 Energy scale ~ meV

Neutrons and complementary techniques